How structures of various scales formed and evolved from the early Universe up to present time is a fundamental question of astrophysics. EDGE will trace the cosmic history of the baryons from the early generations of massive stars by Gamma-Ray Burst (GRB) explosions, through the period of galaxy cluster formation, down to the very low redshift Universe, when between a third and one half of the baryons are expected to reside in cosmic filaments undergoing gravitational collapse by dark matter (the so-called warm hot intragalactic medium). In addition EDGE, with its unprecedented capabilities, will provide key results in many important fields. These scientific goals are feasible with a medium class mission using existing technology combined with innovative instrumental and observational capabilities by: (a) observing with fast reaction Gamma-Ray Bursts with a high spectral resolution (R ~ 500). This enables the study of their (star-forming) environment and the use of GRBs as back lights of large scale cosmological structures; (b) observing and surveying extended sources (galaxy clusters, WHIM) with high sensitivity using two wide field of view X-ray telescopes (one with a high angular resolution and the other with a high spectral resolution). The mission concept includes four main instruments: a Wide-field Spectrometer with excellent energy resolution (3 eV at 0.6 keV), a Wide- Field Imager with high angular resolution (HPD 15") constant over the full 1.4 degree field of view, and a Wide Field Monitor with a FOV of 1/ 4 of the sky, which will trigger the fast repointing to the GRB. Extension of its energy response up to 1 MeV will be achieved with a GRB detector with no imaging capability. This mission is proposed to ESA as part of the Cosmic Vision call. We will briefly review the science drivers and describe in more detail the payload of this mission.